Abstract

For a given Sturm-Liouville equation whose leading coefficient
function changes sign, we establish inequalities among the eigenvalues
for any coupled self-adjoint boundary condition and those for two
corresponding separated self-adjoint boundary conditions. By a recent
result of Binding and Volkmer, the eigenvalues (unbounded from both
below and above) for a separated self-adjoint boundary condition can
be numbered in terms of the Pr\"ufer angle; and our inequalities can
then be used to index the eigenvalues for any coupled self-adjoint
boundary condition. Under this indexing scheme, we determine the
discontinuities of each eigenvalue as a function on the space of such
Sturm-Liouville problems, and its range as a function on the space of
self-adjoint boundary conditions. We also relate this indexing scheme
to the number of zeros of eigenfunctions. In addition, we
characterize the discontinuities of each eigenvalue under a different
indexing scheme.